Exam 4: Estimation: How Large Is the Effect

Researchers are interested in investigating the effect of a drug that is to be used in the treatment of patients who have glaucoma (an eye disorder associated with high eye pressure). In a volunteer sample of 35 patients with glaucoma in both eyes, one eye of each patient was randomly assigned to this drug, and the other eye was given a placebo. After one week, the eye pressure was measured on each eye. The difference in eye pressure between the two eyes (drug - placebo) was measured for each patient. The sample mean difference in eye pressure was -1.21 mmHg (millimeters of mercury), and the sample standard deviation of the differences was 4.67 mmHg. -Suppose a 95% confidence interval for the population mean difference in eye pressure is (L, U). How would you interpret this interval?

Suppose you plan to offer a study session and need to know how many candies to bring and there-fore would like to estimate how many candies students tend to take on average. Below are results for the number of candies that 19 undergraduates took from a large bowl. -Provide an interpretation of the 95% confidence interval given in the output.

Many studies have investigated the question of whether people tend to think of an odd number when they are asked to think of a single-digit number (0 through 9; 0 is considered an even number). When asked to pick a number between 0 and 9, out of 70 students, 42 chose an odd number. Let the parameter of interest, π, represent the probability that a student will choose an odd number. -Are conditions met to use theory-based methods?

A 2012 survey of 1,254 American cell phone owners indicates that 17% of those surveyed do their browsing on their phone rather than a computer or other device. You use these data to calculate a 95% confidence interval for the proportion of all American cell phone owners that do their browsing on their phone rather than a computer or other device. -Suppose you constructed a 90% confidence interval with these data instead. How would your 90% confidence interval compare to your 95% confidence interval? Your 90% confidence interval would be:

A 95% confidence interval for a population proportion is (0.67, 0.83). What is the margin of error?

Suppose a 95% confidence interval for a population proportion is (0.21, 0.43). How would the width of the confidence interval change if we changed the confidence level?

Hill and Barton (Nature, 2005) conducted a study to investigate whether Olympic athletes in certain uniform colors have an advantage over their competitors. They noted that competitors in the combat sports of boxing, Taekwondo, Greco-Roman wrestling, and freestyle wrestling are randomly assigned to wear red or blue uniforms. For each match in the 2004 Olympics, they recorded the uniform color of the winner. Hill and Barton found that in the 457 matches, the competitor wearing red won 248 times (54.3%), whereas the person wearing blue won 209 times (45.7%). -Compute the standard error of the sample proportion of matches in which the winner wore red.

To investigate whether students tend to think of an odd number when they are asked to think of a single-digit number, you used the One Proportion applet and ran a simulation with 100 trials under the null hypothesis. Given below is a dotplot with results from that simulation. Based on this dotplot, would a 95% confidence interval for contain the value 0.5?

The spinning dancer (or silhouette illusion) is a moving image of a woman that appears to be spinning. Some people see her spinning clockwise and some see her spinning counterclockwise. A student showed other students this and found that 30 out of 50 (or 60%) saw her spinning clockwise. The student researcher was interested in the probability of a randomly chosen student that would see the dancer spinning clockwise. He created the following null distribution for this. Using the information provided above, determine a 2SD confidence interval for the long-run probability a randomly chosen student would see the woman spinning clockwise. (Don't use any other applet to do this, just the applet output and information above.)

Researchers are studying a new weight-loss program. Using a sample of 9,812 volunteers with a desire to lose weight, they construct a 95% confidence interval for the true mean amount of weight loss after six months on the program to be (0.12, 0.20) pounds. -Based on this interval, is there statistically significant evidence that the weight-loss program works?

Many studies have investigated the question of whether people tend to think of an odd number when they are asked to think of a single-digit number (0 through 9; 0 is considered an even number). When asked to pick a number between 0 and 9, out of 70 students, 42 chose an odd number. Let the parameter of interest, π, represent the probability that a student will choose an odd number. -Compute the standard error of the sample proportion of students who chose an odd number.

A 95% confidence interval for a population proportion is 0.20 ± 0.09. What are the endpoints of the interval? (______ (1), ______ (2))

As the confidence level decreases, the width of a confidence interval also decreases, all else being equal.

Suppose you plan to offer a study session and need to know how many candies to bring and there-fore would like to estimate how many candies students tend to take on average. Below are results for the number of candies that 19 undergraduates took from a large bowl. -Suppose I wanted to test whether the average number of candies taken was significantly different from 60, the expected number in a small bag of candies. Based on the above output, what conclusion will you draw, or do you not have enough information?

Suppose I am conducting a test of significance with H₀: = 0.50 and H_a: ? 0.50 and I end up with a p-value of 0.07. I also construct 99% and 95% confidence intervals from my data. What will be true about my confidence interval?

Can domestic dogs understand human body cues such as leaning? The experimenter leaned toward one of two objects and recorded whether or not the dog being tested correctly chose the object indicated. A four-year-old male beagle named Augie participated in this study. He chose the correct object 8 out of 10 times when the experimenter leaned towards the correct object. Shown below is a simulation of 1000 sample proportions under the assumption that the long-run probability that Augie chooses correct is 0.50. -Suppose that we repeated the same study with Augie, and this time he chose the correct object 16 out of 20 times. Conjecture how, if at all, the center and the width of a 2SD 95% confidence interval would change with these data, compared to the original 8 out of 10 times. The center of the confidence interval would ______ (1). The width of the confidence interval would ______ (2). Drop-down options for (1) and (2): •increase •decrease •remain the same

Many studies have investigated the question of whether people tend to think of an odd number when they are asked to think of a single-digit number (0 through 9; 0 is considered an even number). When asked to pick a number between 0 and 9, out of 70 students, 42 chose an odd number. Let the parameter of interest, π, represent the probability that a student will choose an odd number. -Use the 2SD method to approximate a 95% confidence interval for . (______ (1), ______ (2))

Suppose you plan to offer a study session and need to know how many candies to bring and there-fore would like to estimate how many candies students tend to take on average. Below are results for the number of candies that 19 undergraduates took from a large bowl. -Are validity conditions met in order to use theory-based methods?

Hill and Barton (Nature, 2005) conducted a study to investigate whether Olympic athletes in certain uniform colors have an advantage over their competitors. They noted that competitors in the combat sports of boxing, Taekwondo, Greco-Roman wrestling, and freestyle wrestling are randomly assigned to wear red or blue uniforms. For each match in the 2004 Olympics, they recorded the uniform color of the winner. Hill and Barton found that in the 457 matches, the competitor wearing red won 248 times (54.3%), whereas the person wearing blue won 209 times (45.7%). -Below are the results of 1000 samples from a process with QUOTE = .50. Using the 2SD method, construct a 95% confidence interval for the long-run proportion of matches where the competitor wearing red wins. (______ (1), ______ (2))

Hill and Barton (Nature, 2005) conducted a study to investigate whether Olympic athletes in certain uniform colors have an advantage over their competitors. They noted that competitors in the combat sports of boxing, Taekwondo, Greco-Roman wrestling, and freestyle wrestling are randomly assigned to wear red or blue uniforms. For each match in the 2004 Olympics, they recorded the uniform color of the winner. Hill and Barton found that in the 457 matches, the competitor wearing red won 248 times (54.3%), whereas the person wearing blue won 209 times (45.7%). -Construct a 99% confidence interval based on these data using Theory-Based Inference applet. (______ (1), ______ (2))